Gauss-Chebyshev Quadrature (Second Kind) Calculator

Enter an integrand f(x) and order n to evaluate the weighted integral ∫[-1,1] f(x)√(1−x²)dx using second-kind Gauss-Chebyshev quadrature. Nodes and weights are shown too.

Input

Enter the integrand f(x) and order n to compute the weighted integral ∫_-1^1 f(x)√(1−x²)dx by second-kind Gauss-Chebyshev quadrature.

Integrand f(x)

e.g. exp(x), 1/(2-x), x^2 + 1. Do not include the √(1−x²) weight. See the notes for supported functions, constants and operators.

Order n

Number of nodes (1–256)

Result

Integral ∫ f(x)√(1−x²) dx

1.7754996892

Interval [-1, 1], weight √(1−x²)

Order n

Nodes

Nodes, weights and function values

Shows the nodes x_i, weights w_i and function values f(x_i).

#	Node x_i	Weight w_i	f(x_i)
1	-0.93969262	0.04083295	0.39074792
2	-0.76604444	0.1442256	0.46484817
3	-0.5	0.26179939	0.60653066
4	-0.17364818	0.33854023	0.84059258
5	0.17364818	0.33854023	1.18963695
6	0.5	0.26179939	1.64872127
7	0.76604444	0.1442256	2.15124005
8	0.93969262	0.04083295	2.55919465

How it works

Second-kind Gauss-Chebyshev quadrature approximates the definite integral ∫_{-1}^{1} f(x)√(1−x²)dx with weight function √(1−x²) on [-1, 1].
For order n the nodes (zeros of the second-kind Chebyshev polynomial U_n) are x_i = cos(iπ/(n+1)) for i = 1, …, n.
The weights are w_i = π/(n+1)·sin²(iπ/(n+1)). All weights are positive and sum to π/2.
The integral approximation is Σ_i w_i · f(x_i). The integrand f(x) is evaluated by a parser, not eval.
This rule integrates any polynomial P(x) of degree up to 2n−1 exactly against the weight, i.e. ∫_{-1}^{1} P(x)√(1−x²)dx.
Use it for integrals whose weight decays like √(1−x²) at the endpoints. Enter the bare f(x) without the √(1−x²) factor, which the rule supplies.

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